On a compact five-manifold, the Stieffel-Whitney number w_2w_3 can be nonzero. An example is the manifold SU(3)/SO(3), and also another example is a CP^2 bundle over a circle where the holonomy is given by the complex conjugation automorphism of CP^2.

My question is whether the Stieffel-Whitney number w_1^2 w_3 can be nonzero on a compact five-manifold. I haven't found either a simple proof that it is zero, or a simple example where it is nonzero.